Rational Function Fan Fair

Sometimes when planning a unit, I browse through the Desmos Activity Builder.  When searching for Rational Functions, I came across Dylan Kane’s Building Rational Functions Activity.  Excellent.  I now had a muse.  Here is what I came up with for college algebra:

I like to gush over my students when they do awesome stuff, and this was no exception.  I love it when my classroom is abuzz with sense-making conversations.  I feel like this activity helped students become more comfortable with the structure of rational functions and how that equation structure is reflected in the graphs. Thanks, Dylan, for inspiring some awesome thinking in my class today.

Watching Solitaire in Silence

Remember Windows Solitaire? I have fond memories playing this fantastic digital distracter with my high school beau on his brand new Gateway computer.  We would take turns striving for success in this card-clicking frenzy, the other watching and waiting patiently for the deck to empty.

But have you ever watched someone play solitaire on the computer?  It is so…what word comes to mind?  Frustrating?  Infuriating?  Aggravating, perhaps?  And why is that?

Check out this screenshot:

What if the player was about to click on that blue, flowery deck of cards…would you be fighting the urge to save them from their potentially game-ending error of failing to move the sequence beginning with the six of spades to its rightful place atop the seven of hearts?  Or would you idly sit by and let them to figure out that solitaire is won by carefully searching for card moves before drawing from the deck?  Would you make any suggestions for improving their game once failure was inevitable?

I think this solitaire analogy is a lot like teaching.   I realized fully today why the “productive struggle” is so hard to sustain and perhaps why teachers so often fall back on traditional methods of delivering information to students:  Watching people struggle without intervening is difficult. Just as it’s natural to want to smooth out the path for our children, it’s also tempting to do the same for our students.  It’s just easier (and so much faster) to zip Maria’s (my daughter) coat or buckle her seat belt or pick up her toys.

As a simple, mathematical example, imagine one of your students is attempting to solve a quadratic equation. They start off like this:

Being the savvy algebra teacher you are, you can anticipate the error that the student is most likely going to make.  You’ve seen it hundreds, if not thousands of times.  Your inner teacher voice might be thinking, “For the love of humanity, Herbie (not your real name), set the dang thing equal to zero!  Quadratic formula!  IT’S GOT A SONG, FOR GOODNESS SAKE!”

Instead, you do not impede their solving and let them continue on their merry, algebraic way.

Re-enter teacher voice in your head, “Now look what you did, Herbie.  You’ve gone and…wait…one of those answers is right.  Great.  Now we’ve really got issues.”

So what do we do about this?  Clearly the student needs some redirection and the teacher’s role is to guide the learning.  But had we intervened during earlier steps, we rob this student of a golden opportunity for brain growth.  Plus, we deprive the rest of the class the chance to learn from the misconception.  Even more, what a fantastic extension we have here:  why did the student get part of the problem correct and part incorrect?

In summary, we deny students the opportunity to learn from mistakes if we  prevent them from making mistakes in the first place.

Related Side Note:  I’m currently reading The Gift of Failure by Jessica Lahey.  Her introduction about her son’s shoelace-tying trials seems strikingly familiar.  And I can use this antedote as a reminder when encountering the zippers, and the seat belts, in addition to quadratic equations.

This is Our Theorem – College Algebra

“We came up with a theorem once at my old school.  The teacher has it in a frame behind his desk.”

This statement from one of my college algebra students made me both elated and sad at the same time.  Thrilled because this is the type of mathematics I believe all students should have the chance to engage in on a regular basis.  Disappointed because this type of discovery happens so infrequently in American mathematics classrooms that the incident warranted a sacred place on the wall of this teacher’s room.

In College Algebra, part of today’s learning objective was to define a polynomial function and determine some key features.  I have the awesome types of students that if I were to write down the surly definition and features of a polynomial function onto the whiteboard, each would follow in lock-step and write it in their notebooks solidifying it’s place among mathematical obscurity.

Today, we were going to break that cycle with something different.

But I needed to know where they were at, so I had them write down what they knew about a polynomial function.

After some discussion and leading questions, we were sure that linear, quadratic, cubic, quartic, x^5, x^6, and so on were all polynomial functions.  Awesome. We weren’t, however, as sure about functions including negative exponents, roots, sin/cos, or algebraic fractions.

What makes this group we are sure about special?  Last week, we spent a considerable amount of time on features of functions including domains, end behavior, intercepts, intervals, symmetry, and turning points.  In their groups, I had them examine the graphs of these alleged “polynomials” through the lens of the features of functions.

Two similarities emerged as significant:

Questions:  Was this true of all polynomial functions?  And if both conditions were not met, could we exclude it from our known polynomial functions?  Hiding my initial excitement, I then had them look at our list of “questionable” functions. For example, did “y = 9 + 1/x” meet each of these two criteria?

Christopher Danielson suggested that my class give this new theorem a name, so we could refer back to it with ease:

“Class. We have found that all polynomials blah blah blah…” [while writing the statement of the theorem on the board.]  In mathematics, when we have an important finding like this, and when all mathematicians have agreed the finding is true, it gets a name.  Sometimes it is named for a person, such as ‘Fermat’s Last Theorem’; sometimes it is named for what it says, as in ‘The Triangle Inequality’.  But that name makes it possible to refer to it going forward. It helps us to remember and to use the thing we figured out. So we need to name our theorem. Who has a name they’d like to suggest?”

Alas, the excitement of naming the theorem will have to wait until tomorrow.

Somewhere between Concrete Sequential and Abstract Random

It occurred to be relatively early in during the trimester this past fall that my college algebra students (generally) have no idea what I mean when I say “quadratic function.”  This isn’t because they have never heard it, learned it, or used it.  But that technical of a term simply has not stuck around in their long term memory.

from youcubed.org and visualpatterns.org

I then had them make posters including how it is growing, what the 10th, 100th, 0th, and -1st cases would look like, table, graph, expression, and relationship to the pattern.  Instead of one large poster, they use 4 smaller pieces of paper and tape them together.  That way each group member can contribute simultaneously.

I noticed:

• It is difficult for students to describe how an irregular shape is growing.
• It is even more difficult for them to describe something abstract like the -1 case.
• Many of them expressed the overall growth as “exponential.”
• Most could easily see the two rectangles formed and determine the dimensions with respect to “n.”

I wondered:

• If they could connect the work with patterns to other quadratics.
• How to have a meaningful discussion around the “exponential growth” issue.

Their homework was to answer similar questions for this pattern from You Cubed’s Week of Inspirational Math:

Spoiler alert: the rule for the pattern is f(n) = (x + 1)^2 or f(n) = x^2 + 2x + 1

So where do we go from here, two days before Winter Break?  My goals are to review some specifics on quadratic functions and simultaneously help the students make connections between different representations.  I know what I must do.  I must channel my inner Triangleman.

[Backstory:  Christopher Danielson and I go way back. At least to 2014. Maybe even 2013.  Seriously though, I strive to organize my college algebra class the way Professor Danielson describes in his blog.  I have picked his brain on more than a few occasions and he is gracious enough to give me advice in certain curricular areas. In short, his philosophy titled “They’ll Need it for Calculus” is the foundation of my College Algebra course. ]

Ok, back to room C118.

Me: Write down everything you know about the function y = (x+1)^2

(Most write down the expanded form, some start to graph, but not many)

Me: What other ways can we represent this function?

Students: Tables! Graphs! Pictures! Words! Patterns! Licorice!

Me: Sweet!  Let’s do all of that, minus the Licorice.

(I give them a few minutes to create a table and a graph.)

I circulate and hand each group a half sheet of paper.

Me: Write down the most important thing on your groups list.

At first I wasn’t really concerned what exactly they wrote down, but how they defended their choice. Then I came across this in all of my classes:

We came up with a pleasing list of attributes of a positive parabola that included vertex placement, end behavior, leading coefficients, and rate of change.

Next up for discussion: Parabolas grow exponentially.

Me: Turn to your partner and tell them whether you agree or disagree with this statement and defend your choice.

Students: Yes, words, words, words.

Me: Ok, now the other partner, say how you know something grows exponentially.

Students: Multiplied every time, more words, blah blah blah.

So we agreed that 2, 4, 8, 16, 32, 64… is an example of something that grows exponentially.

Me: Numerically, how can we tell how something is growing?

Students: (eventually) Rate of Change!

We came up with this table and agreed that these two functions were definitely NOT growing in a similar way.

Now on to helping them understand what it means for something to grow quadratically…

Stringing Students Along

If I’ve done one thing consistently this year, it has been Number Talks in my Probability and Statistics classes.  I have seen students who, at the beginning of the trimester, told me flat out, “I can’t do math in my head.” Now that Trimester 1 is coming to an end, those same kids are volunteering multiple strategies in these mental math challenges.

During the trimester, we started with the dot image below and have moved through the four operations, onto decimals, and even dabbled in fractions and percents.

How many dots are there?  How did you count them?

What’s important to me with these number talks is the visible improvement I saw in my students’ confidence and flexibility with numbers.

I’ve shared before about my experience with number talks and I plan to continue these throughout the rest of the school year.  But at the NCTM Regional conference in Minneapolis a couple of weeks ago, I had the pleasure of attending Pam Harris’s session on Problem Strings.  I found that problem strings are very useful when wanting to elicit certain strategies or move toward generalization of a strategy.

Here are my notes from a problem string I did recently with the same group of students I have been doing number talks with.

I noticed:

• Many students did not use “17 sticks in a pack” to figure out sticks in 10 packs
• Many more strategies than expected were shared to find the number of sticks in 6 packs of gum.
• Most students were able to generalize about number of sticks in n packs.
• Participation increased with the multiple opportunities to volunteer their strategies.
• Students could see relationships between the numbers and find the solution in multiple ways because of that relationship.
• There are many implications of these problem strings in secondary mathematics. In this example, the slope formula can be easily elicited through further exploration of the table we made.

I’ve read all of Pam’s books, but getting to see her present problem strings in person really illuminated how these can be useful in my classroom. Thanks, Pam, for opening my mind to this and letting me fangirl you.  I’m looking forward to doing more of these, including recording them.  Stay tuned.

Nrich-ing New School Year

I have made no secret of my unwavering devotion to Nrich.  This University of Cambridge-based website is a treasure trove of rich tasks.  The best part about this website is the ability to engage students with abstract algebra concepts.  Very infrequently does Nrich apply their problems to any real-world concept.  They let the arithmetic itself set the hook.  And the algebra solidifies the concept.

Example:

As a bonus, now that school is underway, they have many “live” problems which are open for class submission.  I’m hoping that my class might want to submit solutions for the Puzzling Place Value problem.

There are a lot of great problems here that I’ve used in my classes.  The only problem with available new problems is now I want to devote all my time to solving and implementing them!

What a Difference 12 Kids Make

from map.mathshell.org

We’ve entered Spring Trimester and the volatile Minnesota weather is cooperating thus far.  If there’s a silver lining to last year’s Spring suckfest, the lack of warmer weather put off the end-of-the-year slide until closer to May.  I’m not sure we’ll have the same luxury this year.

I teach the same level of Algebra 2 that I did last year but my class sizes are a more manageable 22-24 rather than the monstrosity of 36 I had last year.  I know class size isn’t high on Hattie’s list of influences on student achievement, but providing formative evaluation (something VERY influential, according to Hattie) is much more doable with 20-something rather than 30+.

I’ve left the desks in pods because I’m convinced students interact and collaborate mathematically more often when they have multiple classmates within conversation distance.  I want to switch their groups periodically, if only I could get them to sit in their assigned seat!

One of my go-to resources is the Mathematics Assessment Project. Their lessons are robust, and provide good opportunities for students to have great conversations around the mathematics.  This lesson on investments is no exception.  The main activity is a card sort where students match a principal and interest rate of an investment with a formula, graph, table, and description.  But the everything from the pre-assessment to the closing slide makes students think and share.

Here are the openers of the main lesson:

from map.mathshell.org

from map.mathshell.org

My assumption, not being familiar with this group of kids, was that they’d go right for the obvious – Investment 3 is the odd one out because it has a 10% interest rate and the others have a 5% interest rate.  I underestimated them.  They came up with very creative, thoughtful reasons why each investment could be considered the odd one out.  I really like these questions because all three can be correct, and students have an opportunity to defend multiple answers.

The card sort was also spectacular.  I was able to have great conversations with each group about their thinking. (Yes, that’s much easier to do with 24 rather than 36).  What a difference 12 kids makes.  There is so much to this lesson to love.  If you have a unit on exponential functions, give it a try.  I’d love to hear how it went in your classroom.

Math is Messy. So Are Gender Roles.

I have been absolutely humbled by all of the positive feedback I have received from my previous post.  Thank you to infinity for taking the time to read, write, and share.  I believe that it is our common humanity that makes it possible for us to learn from one another, not necessarily our knowledge of content.  There is so much of my sobriety that goes into my teaching.  It is an incredibly freeing feeling to be able to be honest about that part of my life as I blog.

Rose Eveleth wrote a great piece about the roles that girls find themselves taking on in group work.  In short, Eveleth focuses on acknowledging that girls often self-assign the “recording” role, absolving (and downright excluding) themselves from a problem solving opportunity.  The end result, career-wise, may lead women away from high-profiled positions.   As teachers, it’s easy for us to overlook this discrepancy because girls, generally speaking, are neater and more organized, and may seem like the best fit for the job.  In a related article, Dale Baker does a great job of asking teachers to examine gender preferences that exist in our classrooms in order to help encourage all students to step into the “lime light.”

On Friday, I tried a simple version of this.  First, students were presented this scenario (taken from the Math Forum POW section):

The Student Council at Rahkenrole High School is planning a concert.  They’ve hired the Knox Mountain Boys, a popular local band, for \$340.  A poll among the students has shown that if tickets cost \$5, 140 people will come to the concert.  For every dollar the ticket price goes up, 10 fewer people will come, and for every dollar it goes down, 10 more people will come.

I’ve been a huge fan of the Math Forum, long before I joined Twitter (and got to fangirl Max Ray at TMC14).  The reasons might not seem obvious from this scenario, but kids noticed right away that there was no question asked at the end.  What’s brilliant here is that there is literally an infinite number of questions that we could ask here.  Granted, some questions are more important than others, but I framed the task in a way that elicited what I needed.

I handed out a big white piece of paper to each group of 4 and had them divide the paper up into sections.  This way each person in the group was both the recorder and the problem solver.   I asked them to write down 2 questions they think that I would ask about the scenario and one question (anything) that they would ask.  They identified their group’s most important question and put it up on the whiteboards on the wall.

Low and Behold!  They READ MY MIND! They asked about maximizing profit, income, and people, and also requested modeling equations for each.  The excellence in this scenario (and the Math Forum in general) is that it can be applied to so many levels of math for so many reasons.  For example, most high school kids can make a table and figure out a reasonable answer for the maximization questions, and kids with more know-how can develop mathematical models.

Some great things happened:

1. They knew they needed ONE set of answers in the center of their paper.  This meant they had to communicate the work in their section. The traditional group roles dissipated, and they all had equal stake in solving the problem.
2. They solved the problem in so many different ways.  (Do you remember these types of questions from Algebra 1/2?  I’m sure they have a trendy textbook label that alludes me at the moment. But they are solved by making the variable “number of price increases. Interestingly, very few students solved it that way successfully.)
3. They were messy. And I loved it.  In fact, I made the second class use markers exclusively so that they could not erase.
4. They were uncomfortable leaving some of the questions unanswered.  When I didn’t label certain questions as “bonus” or “extension” they felt that all were necessary to be successful.  My goal was for them to collaborate with ownership in their individual contribution.  I may have gotten more joy out of this part than I should have 🙂

Here are some fun photos of their work:

Nrich – For What It’s Worth

One of my favorite problems (and the one I presented at TMC this year) is What it’s Worth? from Nrich.  To say I “like” this problem would be like saying Sarah Hagan “likes” interactive notebooks.  Clearly an understatement.

Anyway, here’s the prompt:

What I like most about this problem is that there are so many, OH so many, methods to solving it.  It is a FANTASTIC way to get students to focus on the pathways to the solution rather than the solution itself.  After the students figure out the value of the question mark, they go about discussing the numerous methods they used in order to arrive at their answer.  Furthermore, the problem includes 6 “beginnings” of solutions and learners then need to make sense of those as well as determine how a solution was reached along that path.

Left: New format for discerning methods. Right: Old format for discerning methods.

To my surprise, along with Nrich’s site updates, this problem has improved as well.  Rather than showing a written start to the problem, provided are 6 visual introductions.

This allowed for an incredible amount of discussion involving each method.  And even those METHODS broke down into different methods.  It was method madness (awesome madness).

Welding Math and Metal – Day 2

If I needed to choose the most productive portion of most students’ week, Monday morning, first hour would be pretty low on the hierarchy of engagement.  I was undeterred because making sure we had the correct solution was important.

We discussed that the radius of the spool would decrease every time a layer of wire was used.  They began calculating the resulting wire as layers were removed.  This served as an excellent opportunity to introduce summation notation and a great practical use for the mathematics behind it.  It seemed like a much better option than to add up dozens of calculations anyway.

When we arrived at our correct answer (with the desired units) of 1.98 miles, the questions and estimating didn’t end.  They wanted to know how far they could stretch such a wire.  Would it go to the edge of our campus and back?  Would it go from here to the middle school?  Could you go all the way to the grocery store?

They settled on taking the wire, running it out to the edge of the soccer practice fields and then running it all the way to the middle school sign.  It ended up being, to the hundredth, the exact amount of wire we had, provided that someone would stand and hold the wire at the edge of the soccer field.  I loved the attention to precision. I also loved that they were so savvy with Google Earth.